Rational Probability Research Articles

Can a dialetheist stay regular?

For a dialetheist, it is rational to believe that true contradictions exist. However, we argue that a dialetheist faces a complex dilemma given some bridge principles for rational beliefs that connect possibility and probability, including the so-called ‘Regularity Principle’. Either her belief is not doxastically possible even for her, or she must assign positive credence to the proposition that dialetheia exists. The former makes her belief prima facie self-defeating. The latter seems to compel her to choose between several more fine-grained options, each of which seems undesirable for a regular and rational probability theory. It is unclear how a dialetheist would stay rationally regular, so we pose this challenge, seeking a response from a dialetheist.

No Eleventh Conditional Ingleton Inequality

A rational probability distribution on four binary random variables X , Y , Z , U is constructed which satisfies the conditional independence relations [ X ⊥ ⊥ Y ] , [ X ⊥ ⊥ Z | U ] , [ Y ⊥ ⊥ U | Z ] and [ Z ⊥ ⊥ U | X Y ] but whose entropy vector violates the Ingleton inequality. This settles a recent question of Studený (IEEE Trans. Inf. Theory vol. 67, no. 11) and shows that there are, up to symmetry, precisely ten inclusion-minimal sets of conditional independence assumptions on four discrete random variables which make the Ingleton inequality hold. The last case in the classification of which of these inequalities are essentially conditional is also settled.

Outlier detection in a multiset-valued information system based on rough set theory and granular computing

Outlier detection on data with missing information values is especially tricky because the uncertainty caused by missing information values may contribute to an object being an outlier. A multiset-valued information system (MSVIS) is an information system (IS) in which information values are multisets. This kind of IS is a useful way of handling datasets with missing information values. In this paper, we study outlier detection in an MSVIS based on rough set theory and granular computing. First, some concepts of multisets and probability distribution sets are reviewed, and the fact that a weak one-to-one correspondence exists between multisets and rational probability distribution sets is illustrated. In this way, multisets may be treated as rational probability distribution sets. Then, an MSVIS can be induced by an incomplete information system (IIS) and viewed as the result of information fusion of multiple categorical ISs. Next, a tolerance relation in an MSVIS is constructed with the induced rational probability distribution sets. Then, the outlier factor in an MSVIS is formulated, and the corresponding outlier detection algorithm is proposed. Finally, the performance evaluation by AUC (area under the curve) and F1-score shows the superiority of the proposed algorithm over some existing algorithms.

On model robustness of the regime switching approach for pegged foreign exchange markets

We test the robustness of the regime switching model for pegged markets introduced by Drapeau et al. [How rational are the option prices of the Hong Kong dollar exchange rate? J. Derivatives, 2021, 28(3), 140–161]. In particular, there are two disputable underlying assumptions: (1) a Black and Scholes model with low volatility for the pre-depegging regime and (2) a thin tail distribution—Exponential type—for the time of the depegging. For the pre-depegging regime, we consider a bounded model within the peg—from Ingersoll and Rady. For the depegging time, we consider fat tail distributions more in line with catastrophic events—Pareto/Fréchet. We derive the option prices formula for each combination of these models. We then calibrate to option data from USD-HKD as well as EUR-CHF. In comparison to the benchmark model in Drapeau et al. [How rational are the option prices of the Hong Kong dollar exchange rate? J. Derivatives, 2021, 28(3), 140–161], it turns out that the relevant resulting characteristics—probability of a depegging before maturity, appreciation/depreciation at the depegging time as well as post-depegging volatility—are strongly robust in terms of model choice for this regime switching approach. However, from a term structure perspective, fat tail distributions fit the data significantly better and provide more rational depegging probabilities for short and long maturities.

Dual mutations collaboration mechanism with elites guiding and inferiors eliminating techniques for differential evolution

Differential evolution (DE) is a powerful evolutionary algorithm for global optimization problems. Generally, appropriate mutation strategies and proper equilibrium between global exploration and local exploitation are significant to the performance of DE. From this consideration, in this paper, we present a novel DE variant, abbreviated to DMIE-DE, to further enhance the optimization capacity of DE by developing a dual mutations collaboration mechanism with elites guiding and inferiors eliminating techniques. More specifically, an explorative mutation strategy DE/current-to-embest with an elite individual serving as part of the difference vector and an exploitative mutation strategy DE/ebest-to-rand with selecting an elite individual as the base vector are employed simultaneously to achieve the balance between local and global performance of the whole population instead of only one mutation strategy in classical DE algorithm. The control parameters F and CR for above mutation strategies are updated adaptively to supplement the optimization ability of DMIE-DE based on a rational probability distribution model and the successful experience from the previous iterations. Moreover, an inferior solutions eliminating technique is embedded to enhance the convergence speed and compensate cost of the fitness evaluation times during the evaluation process. To evaluate the performance of DMIE-DE, experiments are conducted by comparing with five state-of-the-art DE variants on solving 29 test functions in CEC2017 benchmark set. The experimental results indicate that the performance of DMIE-DE is significantly better than, or at least comparable to the considered DE variants.

Delay analysis of a discrete-time single-server queue with an occasional extra server

In this work we look at the delay analysis of a customer in a discrete-time queueing system with one permanent server and one occasional extra server. The arrival process is assumed to be general independent, the buffer size infinite and the service times deterministically equal to one slot. The system is assumed to be in one of two different states; during the UP-state 2 servers are available and during the DOWN-state 1 server is available. State changes can only occur at slot boundaries and mark the beginnings and ends of UP-periods and DOWN-periods. The lengths of the UP-periods, expressed in their number of slots, are assumed to follow a geometric distribution, while the lengths of the DOWN-periods follow a general distribution with rational probability generating function. We provide a method to compute the tail characteristics of the delay of an arbitrary customer based on the theory of the dominant singularity. We illustrate the developed method with several numerical examples.

Can Shiozawa’s, Morioka’s and Taniuchi’s Microfoundations for Evolutionary Economics (2019) Serve As the Microfoundations for “… Post-Keynesian Economics “ (2019, p.vii)? The Answer Is Definitely Yes if Post –Keynesians Can Break Away From Joan Robinson’s Anti-Mathematical, Anti-Formalist Views

Can Shiozawa’s, Morioka’s and Taniuchi’s Microfoundations for Evolutionary Economics (2019) Serve As the Microfoundations for “… Post-Keynesian Economics “ (2019, p.vii)? The Answer Is Definitely Yes if Post –Keynesians Can Break Away From Joan Robinson’s Anti-Mathematical, Anti-Formalist Views

Analysis of a 2-state discrete-time queue with stochastic state-period lengths and state-dependent server availability and arrivals

In this work we study a discrete-time multiserver queueing system with an infinite storage capacity and deterministic service times equal to 1 slot. Specific to the model under study is that the system is assumed to be in one of two different states (state-1 or state-2) and that both the distribution of the number of available servers and the arrival process depend on the system state. State changes can only occur at slot boundaries and mark the beginning and end of state-1-periods and state-2-periods. The lengths of these state-1-periods and state-2-periods, expressed as a number of slots, are assumed to be two independent sets of independent and identically distributed stochastic variables. The number of available servers during a slot is a stochastic variable with a distribution that is completely determined by the system state during that slot. Likewise, the distribution of the number of arrivals during a slot only depends on the system state during that slot. The only restrictions we put on the distributions of the state-1-periods, state-2-periods and number of available servers is that they have rational probability generating functions (pgfs), and that during each slot at least one server is available. For the considered queueing system we present a method to determine the pgf of the steady-state system content at various observation instants. Several numerical examples demonstrate the possibilities of this model.

Behavioral complexity of British gambling advertising

Background: The scale and complexity of British gambling advertising has increased in recent years. ‘Live-odds’ TV gambling adverts broadcast the odds on very specific, complex, gambles during sporting events (e.g. in soccer, ‘Wayne Rooney to score the first goal, 5-to-1,’ or, ‘Chelsea to win 2-1, 10-to-1’). These gambles were analyzed from a behavioral scientific perspective (the intersection of economics and psychology).Method: A mixed methods design combining observational and experimental data. A content analysis showed that live-odds adverts from two months of televised English Premier League matches were biased towards complex, rather than simple, gambles. Complex gambles were also associated with high bookmaker profit margins. A series of experiments then quantified the rationality of participants’ forecasts across key gambles from the content analysis (Total N = 1467 participants across five Experiments).Results: Soccer fans rarely formed rational probability judgments for the complex events dominating gambling advertising, but were much better at estimating simple events.Conclusions: British gambling advertising is concentrated on the complex products that mislead consumers the most. Behavioral scientific findings are relevant to the active public debate about gambling.

Book review: Review of Particle Filters for Random Set Models

Branko Ristic has given us a superb new book that combines two hot topics (particle filters and random sets) into a delicious feast of algorithms, Monte Carlo simulations, important applications, and mathematics that is new for normal engineers, all served up with clarity and thoroughness. The seminal research on random sets applied to tracking and data fusion problems is due to Ron Mahler [1]. It is clearly a good idea to apply random sets to such problems; in fact, it is hard to avoid using them either implicitly or explicitly. For example, the set of hypotheses for the well-known multiple hypothesis tracking (MHT) algorithms are random sets. The set of hypotheses for MHT is obviously random, because it depends on the measurements (which are random owing to noise errors and random target states). This is exactly the viewpoint adopted in [5], in which one is forced to compute equivalence classes of random sets in order to obtain a set of hypotheses that is mutually exclusive and collectively exhaustive, as required for any rational probability model of the situation (Bayesian or otherwise). It turns out that “fuzzy sets” correspond to equivalence classes of random sets, as explained by Goodman [6]. The set of particles in particle filters is also random, because it is generated from random numbers, but almost no papers explicitly use non-trivial random set theory for particle filter design. In fact, there are very few papers that explicitly talk about random sets and particle filters together, with the conspicuous exception of [20]. Nevertheless, random sets are a mature and well documented mathematical tool, as shown in [1], [2], [3], and [4]. You should not be afraid of random set mathematics; Ron Mahler has written a series of highly accessible tutorials on using random finite sets which avoid the pathologies and complexity of random unfettered sets (e.g., [26] and [28]).

Impact of continuing medical education in cancer diagnosis on GP knowledge, attitude and readiness to investigate - a before-after study.

BackgroundContinuing medical education (CME) in earlier cancer diagnosis was launched in Denmark in 2012 as part of the Danish National Cancer Plan. The CME programme was introduced to improve the recognition among general practitioners (GPs) of symptoms suggestive of cancer and improve the selection of patients requiring urgent investigation. This study aims to explore the effect of CME on GP knowledge about cancer diagnosis, attitude towards own role in cancer detection, self-assessed readiness to investigate and cancer risk assessment of urgently referred patients.MethodsWe conducted a before-after study in the Central Denmark Region including 831 GPs assigned to one of eight geographical clusters. All GPs were invited to participate in the CME at three-week intervals between clusters. A questionnaire focusing on knowledge, attitude and clinical vignettes was sent to each GP one month before and seven months after the CME. The GPs were also asked to assess the risk of cancer in patients urgently referred to a fast-track cancer pathway during an eight-month period. CME-participating GPs were compared with reference (non-participating) GPs by analysing before-after differences.ResultsOne quarter of all GPs participated in the CME. 202 GPs (24.3 %) completed both the baseline and the follow-up questionnaires. 532 GPs (64.0 %) assessed the risk of cancer before the CME and 524 GPs (63.1 %) assessed the risk of cancer after the CME in urgently referred consecutive patients. Compared to the reference group, CME-participating GPs statistically significantly improved their understanding of a rational probability of diagnosing cancer among patients urgently referred for suspected cancer, increased their knowledge of cancer likelihood in a 50-year-old referred patient and lowered the assessed risk of cancer in urgently referred patients.ConclusionsThe standardised CME lowered the GP-assessed cancer risk of urgently referred patients, whereas the effect on knowledge about cancer diagnosis and attitude towards own role in cancer detection was limited. No effect was found on the GPs’ readiness to investigate. CME may be effective for optimising the interpretation of cancer symptoms and thereby improve the selection of patients for urgent cancer referral.Trial registrationNCT02069470 on ClinicalTrials.gov. Retrospectively registered, 1/29/2014.Electronic supplementary materialThe online version of this article (doi:10.1186/s12875-016-0496-x) contains supplementary material, which is available to authorized users.

Identifying the Probability Distribution of Fatigue Life Using the Maximum Entropy Principle

It is well-known that the fatigue lives of materials and structures have a considerable amount of scatter and they are commonly suggested to be considered in engineering design. In order to reduce the introduction of subjective uncertainties and obtain rational probability distributions, a computational method based on the maximum entropy principle is proposed for identifying the probability distribution of fatigue life in this paper. The first four statistical moments of fatigue life are involved to formulate constraints in the maximum entropy principle optimization problem. An accurate algorithm is also presented to find the Lagrange multipliers in the maximum entropy distribution, which can avoid the numerical singularity when solving a system of equations. Two fit indexes are used to measure the goodness-of-fit of the proposed method. The rationality and effectiveness of the proposed method are demonstrated by two groups of fatigue data sets available in the literature. Comparisons among the proposed method, the lognormal distribution and the three-parameter Weibull distribution are also carried out for the investigated groups of fatigue data sets.

Towards a pattern-based logic of probability judgements and logical inclusion “fallacies”

ABSTRACTProbability judgements entail a conjunction fallacy (CF) if a conjunction is estimated to be more probable than one of its conjuncts. In the context of predication of alternative logical hypothesis, Bayesian logic (BL) provides a formalisation of pattern probabilities that renders a class of pattern-based CFs rational. BL predicts a complete system of other logical inclusion fallacies (IFs). A first test of this prediction is investigated here, using transparent tasks with clear set inclusions, varying in observed frequencies only. Experiment 1 uses data where BL makes dominant predictions; Experiment 2's predictions were less clear, and we additionally investigated judgements about the second-most probable hypotheses. The results corroborated a pattern-probability account and cannot be easily explained by other theories of CFs (e.g. inverse probability, confirmation). IFs were not limited to conjunctions, but rather occurred systematically for several logical connectives. Thus, pattern-based probability judgements about logical relations may constitute a basic class of intuitive but potentially rational probability judgements.

Area and Perimeter of the Convex Hull of Stochastic Points

Given a set $P$ of $n$ points in the plane, we study the computation of the probability distribution function of both the area and perimeter of the convex hull of a random subset $S$ of $P$. The random subset $S$ is formed by drawing each point $p$ of $P$ independently with a given rational probability $\pi_p$. For both measures of the convex hull, we show that it is \#P-hard to compute the probability that the measure is at least a given bound $w$. For $\varepsilon\in(0,1)$, we provide an algorithm that runs in $O(n^{6}/\varepsilon)$ time and returns a value that is between the probability that the area is at least $w$, and the probability that the area is at least $(1-\varepsilon)w$. For the perimeter, we show a similar algorithm running in $O(n^{6}/\varepsilon)$ time. Finally, given $\varepsilon,\delta\in(0,1)$ and for any measure, we show an $O(n\log n+ (n/\varepsilon^2)\log(1/\delta))$-time Monte Carlo algorithm that returns a value that, with probability of success at least $1-\delta$, differs at most $\varepsilon$ from the probability that the measure is at least $w$.

Heuristics can produce surprisingly rational probability estimates: Comment on Costello and Watts (2014).

Costello and Watts (2014) present a model assuming that people's knowledge of probabilities adheres to probability theory, but that their probability judgments are perturbed by a random noise in the retrieval from memory. Predictions for the relationships between probability judgments for constituent events and their disjunctions and conjunctions, as well as for sums of such judgments were derived from probability theory. Costello and Watts (2014) report behavioral data showing that subjective probability judgments accord with these predictions. Based on the finding that subjective probability judgments follow probability theory, Costello and Watts (2014) conclude that the results imply that people's probability judgments embody the rules of probability theory and thereby refute theories of heuristic processing. Here, we demonstrate the invalidity of this conclusion by showing that all of the tested predictions follow straightforwardly from an account assuming heuristic probability integration (Nilsson, Winman, Juslin, & Hansson, 2009). We end with a discussion of a number of previous findings that harmonize very poorly with the predictions by the model suggested by Costello and Watts (2014).

Algorithms for Generating Probabilities with Multivalued Stochastic Relay Circuits

The problem of random number generation dates back to Von Neumann’s work in 1951. Since then, many algorithms have been developed for generating unbiased bits from complex correlated sources as well as for generating arbitrary distributions from unbiased bits. An equally interesting, but less studied aspect is the structural component of random number generation. That is, given a set of primitive sources of randomness, and given composition rules induced by a device or nature, how can we build networks that generate arbitrary probability distributions? In this paper, we study the generation of arbitrary probability distributions in multivalued relay circuits, a generalization in which relays can take on any of $N$ states and the logical ‘and’ and ‘or’ are replaced with ‘min’ and ‘max’ respectively. These circuits can be thought of as modeling the timing of events which depend on other event occurences. We describe a duality property and give algorithms that synthesize arbitrary rational probability distributions. We prove that these networks are robust to errors and design a universal probability generator which takes input bits and outputs any desired binary probability distribution.

Sources of uncertainty in liquefaction triggering procedures

ABSTRACTUncertainties in observed data and in processing field and laboratory tests are major concerns. Assigning reasonable coefficients of variation to the parameters in the conventional analyses indicates that a site with deterministic factors of safety of 1.5 can actually have liquefaction triggering probability above 20%. About a third of the variance comes from uncertainty in the load, which is independent of the resistance. Researchers have traditionally presented the results of case studies in the form of charts showing instances in which liquefaction did and did not occur and have developed relations to separate the two. Although the original researchers developed the separations informally, recent work has applied statistical methods. These give the sampling distributions of the observed data rather than the probability of triggering given the data. Researchers have addressed this issue using Bayesian methods, adopting non-informative priors. Published curves of liquefaction probabilities can be interpreted as likelihood ratios. Other independent work demonstrates that geological, meteorological, and historical data can be used to develop prior probabilities, so it may not be necessary to assume a non-informative prior. The actual prior can then be combined with the likelihood ratios to provide rational probabilities of liquefaction. We recommend that researchers publish their likelihood ratios and allow engineers faced with particular sites to use those to update their own priors.

Intime conviction et doute. Au-delà d’une antinomie apparente… Ce que révèle leur étude comparée

The “intimate conviction” is analysed as a moral certainty, which should be incompatible with any doubt in the mind of the judge. In fact, the doubt cannot be entirely excluded, so that the “intimate conviction” is a way to overcome doubt, allowing the judge to take his decision. The “intimate conviction”, however, is a result of a “dialectical” logic based on rational probabilities but does not lead to an absolute truth. This is the reason why the judges are required to give reasons. The “Cour de cassation” controls the duty to state reasons for judicial decisions, in order to ensure their rationality. But case law does not seem very strict about the convictions, unlike the acquittals for the benefit of the doubt. Therefore, as in English law, the “intimate conviction” must be achieved from the time where guilt is demonstrated, in a rational manner, “on beyond a reasonable doubt”.

Statistical Data and Mathematical Propositions

AbstractStatistical tests of the primality of some numbers look similar to statistical tests of many nonmathematical, clearly empirical propositions. Yet interpretations of probability prima facie appear to preclude the possibility of statistical tests of mathematical propositions. For example, it is hard to understand how the statement that n is prime could have a frequentist probability other than 0 or 1. On the other hand, subjectivist approaches appear to be saddled with ‘coherence’ constraints on rational probabilities that require rational agents to assign extremal probabilities to logical and mathematical propositions. In the light of these problems, many philosophers have come to think that there must be some way to generalize a Bayesian statistical account. In this article I propose that a classical frequentist approach should be reconsidered. I conclude that we can give a conditional justification of statistical testing of at least some mathematical hypotheses: if statistical tests provide us with reasons to believe or bet on empirical hypotheses in the standard situations, then they also provide us with reasons to believe or bet on mathematical hypotheses in the structurally similar mathematical cases.

Synthesis of Stochastic Flow Networks

A stochastic flow network is a directed graph with incoming edges (inputs) and outgoing edges (outputs), tokens enter through the input edges, travel stochastically in the network, and can exit the network through the output edges. Each node in the network is a splitter, namely, a token can enter a node through an incoming edge and exit on one of the output edges according to a predefined probability distribution. Stochastic flow networks can be easily implemented by DNA-based chemical reactions, with promising applications in molecular computing and stochastic computing. In this paper, we address a fundamental synthesis question: Given a finite set of possible splitters and an arbitrary rational probability distribution, design a stochastic flow network, such that every token that enters the input edge will exit the outputs with the prescribed probability distribution. The problem of probability transformation dates back to von Neumann's 1951 work and was followed, among others, by Knuth and Yao in 1976. Most existing works have been focusing on the "simulation" of target distributions. In this paper, we design optimal-sized stochastic flow networks for "synthesizing" target distributions. It shows that when each splitter has two outgoing edges and is unbiased, an arbitrary rational probability \frac{a}{b} with a\leq b\leq 2^n can be realized by a stochastic flow network of size n that is optimal. Compared to the other stochastic systems, feedback (cycles in networks) strongly improves the expressibility of stochastic flow networks.